We first come to study the four-fold degenerate nodal line phonons along the S-R path. KCuS hosts three orthogonal two-fold screw rotation axes. Considering a combined antiunitary operation \(\cal {T}\)\(\widetilde{C_{2i}}\) ,\((i=x,y,z),\)  one can easily derive that \((\cal {T}\)\(\widetilde{C_{2\textit {i}}})^2=e^{\textit {ik}_{\textit {i}}}.\)  Consequently, at the corresponding plane, \(k_i=\pi ,\)  one has \((\cal {T}\)\(\widetilde{C_{2\textit {i}}})^2=-1.\)  That is, the phonon dispersions along all boundary planes \((k_{x/y/z}=\pi )\)  are at least two-fold degenerate. Therefore, the whole \(k_{x/y/z}=\pi \)  planes are covered by nodal surface phonons [1]}. As shown in Fig. REF (b), one can see that there are two double degenerate nodal lines along Y-T-S, and these two nodal lines are formed into one four-fold degenerate nodal line along the S-R path. Hence, the four-fold degenerate DNL (e.g., S-R path) is sitting at the hinge between the \(k_x=\pi \)  and \(k_y=\pi \)  planes. That is, the phononic DNL (e.g., S-R path) is formed by the crossing of the phononic nodal surface states of \(k_x=\pi \)  and \(k_y=\pi \)  planes. Note that the DNL phonons along S-R belong to open nodal line states [2]}, [1]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, as shown in Fig. REF (c). More interestingly, as shown in Fig. REF (c) and Fig. REF (d), we selected a series of phonon band crossing points (E, F, G, and H points) along the S-R path and showed the phonon dispersions along the A–E–A\(^{\prime }\) , B–F–B\(^{\prime }\) , C–G–C\(^{\prime }\) , and D–H–D\(^{\prime }\)  paths. Obviously, it can be concluded that the DNL along the S-R path is formed by a series of linear phonon band crossing points. These phonon band crossing points are nearly flat with small frequency variation (see the orange circles in Fig. REF (d); the orange circles are almost flat in frequency), and exhibit a large linear frequency range.
